A mixture-model approach for parallel testing for unequal variances

Testing for unequal variances is usually performed in order to check the validity of the assumptions that underlie standard tests for differences between means (the t-test and anova). However, existing methods for testing for unequal variances (Levene's test and Bartlett's test) are notoriously non-robust to normality assumptions, especially for small sample sizes. Moreover, although these methods were designed to deal with one hypothesis at a time, modern applications (such as to microarrays and fMRI experiments) often involve parallel testing over a large number of levels (genes or voxels). Moreover, in these settings a shift in variance may be biologically relevant, perhaps even more so than a change in the mean. This paper proposes a parsimonious model for parallel testing of the equal variance hypothesis. It is designed to work well when the number of tests is large; typically much larger than the sample sizes. The tests are implemented using an empirical Bayes estimation procedure which `borrows information' across levels. The method is shown to be quite robust to deviations from normality, and to substantially increase the power to detect differences in variance over the more traditional approaches even when the normality assumption is valid.     

Comparison of SANOVA Procedure of Simultaneous Testing Hypotheses to Others

In this paper a procedure SANOVA of simultaneous testing hypotheses is compared with others used in analysis of variance in a fixed linear model. The geometrical relation between SANOVA and Scheffé's confidence regions is discussed. It is shown that individual confidence intervals from SANOVA procedure are not longer than Scheffe's, Dunnett's and Tukey's ones. The cases, when they are the same are indicated. Theoretical considerations are illustrated by a practical example.     

Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

On Testing for Non-Additivity in Factorial Experiments

In analysis of variance the assumption of non-additivity has more serious consequences on the interpretation of results than the assumptions of non-normality or non-homogeneity of variances. Tukey (1949, 1955) suggested a procedure for separating out a single degree of freedom for non-additivity for two factor and multi-factor experiments, respectively. In multifactor experiments Tukey's procedure does not indicate which combinations of the factors contribute to non-additivity. In this paper a procedure is proposed which separates out a sum of squares with a single degree of freedom from each of the 2^P?P?1 interactions in a P-factor experiment.     

Robustness of statistical tests for multiplicative terms in the additive main effects and multiplicative interaction model for cultivar trials

The additive main effects multiplicative interaction model is frequently used in the analysis of multilocation trials. In the analysis of such data it is of interest to decide how many of the multiplicative interaction terms are significant. Several tests for this task are available, all of which assume that errors are normally distributed with a common variance. This paper investigates the robustness of several tests (Gollob, F _GH1, F_GH2, F_R)to departures from these assumptions. It is concluded that, because of its better robustness, the F _Rtest is preferable. If the other tests are to be used, preliminary tests for the validity of assumptions should be performed.     

Elimination of inbreeding depression from a captive population of Speke's gazelle: Validity of the original statistical analysis and confirmation by permutation testing

The Speke's gazelle captive breeding program was designed in the early 1980s to simultaneously maintain the population's genetic diversity while reducing the severity of the inbreeding depression in a situation in which inbreeding could not be avoided. Statistical analyses of the resulting data using both regression techniques and nonparametric exact contingency tests revealed that the inbreeding depression was indeed reduced, and genetic surveys revealed that high levels of nuclear genetic diversity had indeed been maintained. Hence, the twin goals of the breeding program appeared to have been achieved. Recently, several papers have been published that question the validity of the original statistical analyses and resulting biological conclusions. Specifically, these papers raise three major issues: (1) that a small sample correction factor used in the regression analysis represents a statistical “flaw,” (2) that new analyses of the data do not confirm the original conclusion of a significant reduction in the level of inbreeding depression, and (3) that the biological conclusions about the program are not justified. In this paper we show (1) that there is no “flaw” in the small sample correction, (2) that the recent permutational test given by Willis and Wiese seriously violates standard procedures and has no statistical validity, (3) that the regression procedures used by Ballou are inappropriate because the data seriously violate the underlying statistical assumptions and that the statistically valid components of Ballou's work strongly confirm the validity of the Speke's gazelle program, (4) that permutational tests done in accordance with standard statistical practice strongly confirm the results of the original analysis, and (5) that the original biological conclusions are fully justified by multiple types of statistical analyses. Zoo Biol 17:77–94, 1998. © 1998 Wiley-Liss, Inc.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

Maximising sensitivity for detecting changes in protein expression: experimental design using minimal CyDyes

DIGE is a powerful tool for measuring changes in protein expression between samples. Here we assess the assumptions of normality and heterogeneity of variance that underlie the univariate statistical tests routinely used to detect proteins with expression changes. Furthermore, the technical variance experienced in a multigel experiment is assessed here and found to be reproducible within- and across-sample types. Utilising the technical variance measured, a power study is completed for several "typical" fold changes in expression commonly used as thresholds by researchers. Based on this study using DeCyder, guidance is given on the number of gel replicates that are needed for the experiment to have sufficient sensitivity to detect expression changes. A two-dye system based on utilising just Cy3 and Cy5 was found to be more reproducible than the three-dye system. A power and cost-benefit analysis performed here suggests that the traditional three-dye system would use fewer resources in studies where multiple samples are compared. Technical variance was shown to encompass both experimental and analytical noise and thus is dependent on the analytical software utilised. Data is provided as a resource to the community to assess alternative software and upgrades.     

The Effect of Categorization on Balanced Incomplete Blocks and Latin Squares

Robustness in the various analysis of variance models should be assessed with regard to all the assumptions. Here we assess the effect of the continuity assumption. The models considered were chosen because they are in a sense fragile, and the effects might be anticipated to be more dramatic here. We identify at least one of the balanced incomplete block models for which severe categorization leads to tests with unacceptable sizes and aberrant power curves. Otherwise most of the models examined are generally satisfactory – that is, robust with respect to the continuity assumption.     

A Note Concerning “Some Nonparametric Tests of Predicted Order”

SARRIS and WILKENING (1977) have recently proposed some non-parametric trend tests, which they view as extensions of MOSTELLER 'S test of predicted order. The present paper notes some errors in SARRIS and WILKENING 's determination of significance levels for these tests, and describes how these errors may be corrected.     

Testing the metabolic theory of ecology

The metabolic theory of ecology (MTE) predicts the effects of body size and temperature on metabolism through considerations of vascular distribution networks and biochemical kinetics. MTE has also been extended to characterise processes from cellular to global levels. MTE has generated both enthusiasm and controversy across a broad range of research areas. However, most efforts that claim to validate or invalidate MTE have focused on testing predictions. We argue that critical evaluation of MTE also requires strong tests of both its theoretical foundations and simplifying assumptions. To this end, we synthesise available information and find that MTE's original derivations require additional assumptions to obtain the full scope of attendant predictions. Moreover, although some of MTE's simplifying assumptions are well supported by data, others are inconsistent with empirical tests and even more remain untested. Further, although many predictions are empirically supported on average, work remains to explain the often large variability in data. We suggest that greater effort be focused on evaluating MTE's underlying theory and simplifying assumptions to help delineate the scope of MTE, generate new theory and shed light on fundamental aspects of biological form and function.     

Tables and Applications of the Bonferroni t-Statistics: A Revision of Dunn's Simultaneous t-Tests

Tables for Bonferroni-adjusted significance levels of Student's t are provided for r = 3(1)30(5)50 etc. and alphas of 0.05, 0.01 and 0.001. The tables are suggested for various applications, for example in replacing analysis of variance of k samples by r simultaneous t tests. Use of the tables is shown by numerical examples. The adjustment of alpha for improving the efficiency of testing is made by WILKINSON for orthogonal comparisons and by HOLM for nonorthogonal comparisons.     

Poisson's ratio of bovine meniscus determined combining unconfined and confined compression

Poisson's ratio has not been experimentally measured earlier for meniscus in compression. It is however an important intrinsic material property needed in biomechanical analysis and computational models. In this study, equilibrium Poisson's ratio of bovine meniscus (n = 6) was determined experimentally by combining stress-relaxation measurements in unconfined and confined compression geometries. The average Young's modulus, aggregate modulus and Poisson's ratio were 0.182 ± 0.086 MPa, 0.252 ± 0.089 MPa and 0.316 ± 0.040, respectively. These moduli are consistent with previously determined values, but the Poisson’s ratio is higher than determined earlier for meniscus in compression through biomechanical modelling analysis. This new experimentally determined Poisson’s ratio value could be used in the analysis of biomechanical data as well as in computational finite element analysis when the Poisson’s ratio is needed as an input for the analysis.     

Morphological variations in a widely distributed Eastern Asian passerine cannot be consistently explained by ecogeographic rules

Ecogeographic rules that describe quantitative relationships between morphologies and climate might help us predict how morphometrics of animals was shaped by local temperature or humidity. Although the ecogeographic rules had been widely tested in animals of Europe and North America, they had not been fully validated for species in regions that are less studied. Here, we investigate the morphometric variation of a widely distributed East Asian passerine, the vinous‐throated parrotbill (Sinosuthora webbiana), to test whether its morphological variation conforms to the prediction of Bergmann''s rule, Allen''s rules, and Gloger''s rule. We at first described the climatic niche of S. webbiana from occurrence records (n = 7838) and specimen records (n = 290). The results of analysis of covariance (ANCOVA) suggested that the plumage coloration of these parrotbills was darker in wetter/warmer environments following Gloger''s rule. However, their appendage size (culmen length, beak volume, tarsi length) was larger in colder environments, the opposite of the predictions of Allen''s rule. Similarly, their body size (wing length) was larger in warmer environments, the opposite of the predictions of Bergmann''s rule. Such disconformity to both Bergmann''s rule and Allen''s rule suggests that the evolution of morphological variations is likely governed by multiple selection forces rather than dominated by thermoregulation. Our results suggest that these ecogeographic rules should be validated prior to forecasting biological responses to climate change especially for species in less‐studied regions.     

具有Allee效应的合作进化

Allee效应对物种的续存是潜在的干扰因素,在很大程度上将增加种群局部甚至全局灭绝的可能性。对许多物种,尤其是濒临物种更容易受其影响。将Allee效应引入囚徒困境博弈模型,通过理论分析与数值模拟相结合的方法分析讨论了Allee效应对合作进化的影响。研究结果表明:在恶劣的环境条件下,Allee效应极易使物种灭绝,不利于合作进化;在相对优越的环境条件下(死亡率较低),Allee效应促进合作进化,且Allee效应强度越强,更有利于合作进化,不过种群的空间斑块占有率也会随着Allee效应强度的增强而降低,使物种最终灭绝。     

Reconciling ecogeographical rules: rainfall and temperature predict global colour variation in the largest bird radiation

Ecogeographical rules that associate climate with organismal form and function can reveal patterns of climatic adaptation. Two rules link animal coloration with climate: Gloger's rule (darker coloration where wet and warm), and Bogert's rule (darker coloration where cold). Whereas Gloger's rule was proposed for endotherms, and Bogert's rule for ectotherms, both rules may apply more broadly, despite their seemingly opposing effects. Here, we test this contradiction on a global scale across passerine birds. Consistent with Gloger's rule, birds were darker in wetter areas and, following Bogert's rule, lighter where warm, although birds became lighter again at very low temperatures. Rainfall and temperature had antagonistic or additive effects depending on their pattern of covariation, and this predicted whether birds followed the rules. We integrate both rules into a general framework to explain heterogeneity in climatic effects on coloration, which has implications to understand patterns of diversification, climatic adaptation and climate change impacts.     

Complex segregation analysis of Carabelli's trait in a melanesian population

A complex segregation analysis was performed on Carabelli's trait on the upper first molar utilizing 358 nuclear families from the Solomon Islands of Bougainville and Malaita. Simultaneous estiamtion of three sources of variation by the method of maximum likelihood demonstrates a significant effect of shared sibling environment which accounts for over 19% of the variance in liability for the trait. In addition, a statistically significant major gene influence is discussed and suggestions for quantifying individual liability levels for this and other dental traits are outlined.     

A Comparative Statistical Analysis of Genetic Distances. I - Estimation of Distances and of their Variances: Distribution of the Estimators

We present a Monte-Carlo simulation analysis of the statistical properties of absolute genetic distance and of Nei's minimum and standard genetic distances. The estimation of distances (bias) and of their variances is analysed as well as the distributions of distance and variance estimators, taking into account both gamete and locus samplings. Both of Nei's statistics are non-linear when distances are small and consequently the distributions of their estimators are extremely asymmetrical. It is difficult to find theoretical laws that fit such asymmetrical distributions. Absolute genetic distance is linear and its distributions are better fit by a normal distribution. When distances are medium or large, minimum distance and absolute distance distributions are close to a normal distribution, but those of the standard distance can never be considered as normal. For large distances the jack-knife estimator of the standard distance variance is bad; another standard distance estimator is suggested. Absolute distance, which has the best mathematical properties, is particularly interesting for small distances if the gamete sample size is large, even when the number of loci is small. When both distance and gamete sample size are small, this statistic is biased.     

Greater opportunities for sexual selection in male than in female obligate brood parasitic birds

Females are expected to have evolved to be more discriminatory in mate choice than males as a result of greater reproductive investment into larger gametes (eggs vs. sperm). In turn, males are predicted to be more promiscuous than females, showing both a larger variance in the number of mates and a greater increase in reproductive success with more mates, yielding more intense sexual selection on males vs. females (Bateman's Paradigm). However, sex differences in costly parental care strategies can either reinforce or counteract the initial asymmetry in reproductive investment, which may be one cause for some studies failing to conform with predictions of Bateman's Paradigm. For example, in many bird species with small female‐biased initial investment but extensive biparental care, both sexes should be subject to similar strengths of sexual selection because males and females are similarly restricted in their ability to pursue additional mates. Unlike 99% of avian species, however, obligate brood parasitic birds lack any parental care in either sex, predicting a conformation to Bateman's Paradigm. Here we use microsatellite genotyping to demonstrate that in brood parasitic brown‐headed cowbirds (Molothrus ater), per capita annual reproductive success increases with the number of mates in males, but not in females. Furthermore, also as predicted, the variance of the number of mates and offspring is greater in males than in females. Thus, contrary to previous findings in this species, our results conform to predictions of the Bateman's Paradigm for taxa without parental care.